I've been working for some time on Schimmerling's A course on Set Theory, and, thanks to you guys, I'm now almost finishing chapter 3 on ordinals (hah!). In one of the last exercises, he ask us to find the two functions $f, g$ which obey the following conditions: $f: \omega \to \omega + \omega$, $g: \omega + \omega \to \omega + \omega + \omega$, $\mathrm{sup}(f[\omega]) = \omega + \omega$, $\mathrm{sup}(g[\omega + \omega]) = \omega + \omega + \omega$ and $\mathrm{sup}(g \circ f[\omega]) < \omega + \omega + \omega$.
I thought a little bit about this but couldn't think of functions other than ordinal addition, which (I think) won't give the desired result. If someone could give me a hint in the right direction, I'd appreciate.
HINT: Take $f$ to be a function whose range does not include any infinite intervals; and then pick $g$ to be constant on the range of $f$ and increasing on its complement.